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Fuzzy Stochastic Optimal Guaranteed Cost Control of Bio-Economic Singular Markovian Jump Systems Li Li, Qingling Zhang, and Baoyan Zhu
Abstract—This paper establishes a bio-economic singular Markovian jump model by considering the price of the commodity as a Markov chain. The controller is designed for this system such that its biomass achieves the specified range with the least cost in a finite-time. Firstly, this system is described by Takagi–Sugeno fuzzy model. Secondly, a new design method of fuzzy state-feedback controllers is presented to ensure not only the regularity, nonimpulse, and stochastic singular finitetime boundedness of this kind of systems, but also an upper bound achieved for the cost function in the form of strict linear matrix inequalities. Finally, two examples including a practical example of eel seedling breeding are given to illustrate the merit and usability of the approach proposed in this paper. Index Terms—Bio-economic model, finite-time control, guaranteed cost, linear matrix inequality (LMI), singular Markovian jump system (SMJS), Takagi–Sugeno (T-S) fuzzy model.
I. I NTRODUCTION T PRESENT, humankind is facing the problem of a shortage of resources and a worsening environment. So there has been a rapidly growing interest in the analysis and modeling of biological systems. It is well known that harvesting has a strong impact on dynamic evolution of a population. Nowadays, the biological resources in the preypredator ecosystem are commercially harvested and sold with the aim of achieving economic interest. Furthermore, the harvest effort is usually influenced by the variation of economic interest of harvesting. By considering the economic interest of the harvest effort, a kind of bio-economic models is proposed, which is established by several differential equations and an algebraic equation [1]–[3]. Among them, the algebraic equation is established to investigate the effect of harvest effort
A
Manuscript received March 13, 2014; revised July 25, 2014, October 27, 2014, and November 10, 2014; accepted November 12, 2014. This work was supported by the National Natural Science Foundation of China under Grant 61304054, Grant 61273003, and Grant 61273008. This paper was recommended by Associate Editor P. Shi. (Corresponding author: Qingling Zhang.) L. Li is with the Institute of Systems Science, Northeastern University, Shenyang 110819, China, and also with the College of Mathematics and Physics, Bohai University, Jinzhou 121013, China. Q. Zhang is with the Institute of Systems Science, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China (e-mail:
[email protected]). B. Zhu is with the College of Science, Shenyang Jianzhu University, Shenyang 110168, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2375957
on some population in the ecosystem based on the economic theory proposed in [4]. Differential-algebraic system is also referred to as singular system, descriptor system, or implicit system. Compared with the models proposed in [5]–[7], the advantages of the bio-economic models proposed in [1]–[3] are that these models not only investigate the interaction mechanism of the prey-predator system but also offer a simpler way to study the effect of harvest effort on the ecosystem from an economic perspective. Markovian jump systems have been attracting increasing attention in the past few decades. This kind of systems is very powerful and appropriate to model systems which are subject to abrupt changes in their structure and parameters [8]–[11]. When bio-economic models suffer abrupt changes in their structures such as sudden environmental changes and changing subsystem interconnections, it is natural to model them as bio-economic singular Markovian jump systems (SMJSs). Nowadays, there has been an increasing interest in the study of Takagi–Sugeno (T-S) fuzzy systems because it is a convenient tool to deal with the analysis and synthesis for complex nonlinear systems. Many issues related to the stability analysis and control synthesis for this kind of systems have been studied [12]–[16]. But very few results are concerned on the fuzzy-mode-based technique used to deal with nonlinear SMJSs [17], not to mention this kind of problems with practical background. So, using the method of control for T-S fuzzy SMJSs to solve bio-economic problems will be a new attempt in the field of economic control. On the other hand, large values of the state in a finite-time for some systems are not acceptable owing to the presence of saturations [18], [19]. In order to deal with these transient performances of controlled dynamic systems, finitetime stability or short-time stability was introduced. Some appealing results were obtained to ensure finite-time stability, finite-time boundedness, and finite-time stabilization of various systems including linear systems, nonlinear systems, stochastic systems, and singular systems [20]–[23]. Moreover, the guaranteed cost control was extendedly investigated to stabilize the controlled systems while providing an upper bound on a given performance index by many scholars in the last decade [24]–[26]. For a biological system, due to limited space capacity, large values of population are obviously not acceptable. So it is significant to study the finite-time stability of a bio-economic system. In addition, how to use the minimum
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cost to obtain the biggest profit, while maintaining the sustainable development of the bio-economic system is a topic worthy of study. At the same time, by considering the price of the commodity as a Markovian process, a new bio-economic SMJS is established, which is differing from the traditional research on bio-economic systems. Therefore, stochastic optimal guaranteed cost control of bio-economic SMJSs is studied in this paper. The rest of this paper is organized as follows. In Section II, a bio-economic Markovian jump model is established, and some preliminaries are introduced. The results on stochastic singular finite-time optimal guaranteed cost control are given in Section III. The main contribute lies in a fuzzy state feedback controller designed in the form of linear matrix inequality (LMI), which ensures stochastic singular finite-time boundedness (SSFTB) of the closed-loop system and an upper bound achieved for the cost function. Section IV presents two examples to demonstrate the validity of the proposed methodology. The conclusion is drawn in Section V. A. Notations Throughout this paper, the notations used are fairly standard, for real symmetric matrices A and B, the notation A ≥ B (A > B) means that the matrix A − B is positive semi-definite (positive definite). AT represents the transpose of a matrix A, and A−1 represents the inverse of a matrix A. λmax B (λmin B) is the maximal (minimal) eigenvalue of a matrix B. diag {· · · } stands for a block-diagonal matrix. I is the unit matrix with appropriate dimensions, and in a matrix, the term of symmetry is stated by the asterisk ∗. Let Rn stands for the n-dimensional Euclidean space, Rn×m is the set of all n × m real matrices, and · denotes the Euclidean norm of vectors. E{·} denotes the mathematics expectation of the stochastic process or vector. L2n [0, ∞) stands for the space of n-dimensional square integrable functions on [0, ∞). II. M ODELING AND P RELIMINARY In the natural world, there are many species whose individuals have a life history that takes them through two stages: 1) juvenile stage and 2) adult stage. Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, there is a strong interaction relationship between the mature population and the immature population, which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population. A model describing the dynamics of a single species with stage structure is proposed in [27] can be expressed as follows: x˙ 1 (t) = αx2 (t) − γ1 x1 (t) − βx1 (t) − ηx12 (t) (1) x˙ 2 (t) = βx1 (t) − γ2 x2 (t) where x1 (t) and x2 (t) represent population density of immature species and mature species, respectively; α, γ1 , β denote the intrinsic growth rate, death rate, and transition rate of the
immature population, respectively; γ2 is the death rate of the mature population, and the growth of the immature population is restricted by population density that is reflected by −ηx12 (t). Considering the effect of harvest effort on the ecosystem from an economic perspective, Zhang et al. [3] established a bio-economic singular model system by considering the economic interest of harvest effort on the immature population ⎧ x˙ (t) = αx2 (t) − γ1 x1 (t) − βx1 (t) − ηx12 (t) ⎪ ⎪ ⎨ 1 − (t) x1 (t) = βx x ˙ (t) ⎪ 2 1 (t) − γ2 x2 (t) ⎪ ⎩ 0 = (t) (ρx1 (t) − c) − m
(2)
where (t) is the harvest effort on the immature population, ρ is a price coefficient per the individual population, c is the cost coefficient, c(t) is the total cost, and m is the economic interest of harvesting. It can be seen that (2) is a singular system. When the price coefficient ρ in (2) is considered to be influenced by a Markovian process [28] with right continuous trajectories taking values in a finite set S = {1, 2, . . . , N}
with transition rate matrix = πpq given by Pr {rt+h = q |rt = p } =
p = q πpq h + o (h) 1 + πpp h + o (h) p = q
(3)
where h > 0, lim o(h) h = 0 and πpq ≥ 0, for q = p, is h→0 the transition rate from mode p to q at time t + h, which satisfies πpp = − N q=1,q=p πpq , a bio-economic SMJS can be established here on the basis of model system (2) ⎧ x˙ (t) = αx2 (t) − γ1 x1 (t) − βx1 (t) − ηx12 (t) ⎪ ⎪ ⎨ 1 − (t) x1 (t) = βx x ˙ (t) ⎪ ⎪ 1 (t) − γ2 x2 (t) ⎩ 2 0 = (t) ρrt x1 (t) − c − m.
(4)
According to the economic theory of a common-property resource [4], there is a phenomenon of bio-economic equilibrium when the economic interest of harvesting is zero (m = 0), i.e., total revenue is equal to total cost, which is also known as the economic overfishing. Then, this will cause the ecological imbalance, there must be an ecological disaster. But some artificial means can be used to control it. For example, in order to protect resources and promote the economic development by adjusting the amount of the tax to increase or decrease the cost for harvesting, the government implements control to the bio-economic system such that the system can develop continuously and continue to profit. For model system (4), when m = 0, an interior positive equilibrium P∗ ((c/ρrt ), (cβ/ρrt γ2 ), ((αβρrt − ρrt γ1 γ2 − βρrt γ2 − ηcγ2 )/ρrt γ2 )) can been obtained in the case of the phenomenon of bio-economic equilibrium, where αβρrt − ρrt γ1 γ2 − βρrt γ2 − ηcγ2 > 0. Let z1 (t) = x1 (t) − (c/ρrt ), z2 (t) = x2 (t) − (cβ/ρrt γ2 ), z3 (t) = (t) − ((αβρrt − ρrt γ1 γ2 − βρrt γ2 − ηcγ2 )/ρrt γ2 ). By using the above transformation, the following controlled system with
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external perturbation can be obtained: ⎧
αβ ηc c ⎪ z ˙ − = − (t) ⎪ 1 γ2 ρrt z1 (t) + αz2 (t) − ρrt z3 (t) ⎪ ⎪ ⎪ ⎪ − ηz21 (t) − z1 (t) z3 (t) + b11 w˜ (t) ⎨ z˙2 (t) = βz1 (t) − γ2 z2 (t) +b12 w˜ (t) ⎪ ⎪ ηc ⎪ ⎪ 0 = ρrt αβ ⎪ γ2 − γ1 − β − ρrt z1 (t) ⎪ ⎩ + ρrt z1 (t) z3 (t) + b13 w˜ (t) + u˜ (t)
(5)
where u˜ (t) is the control input representing regulation control for a biological resource, such as the increase or decrease in tax; w(t) ˜ represents perturbation input; b11 , b12 , and b13 denote the perturbation coefficients. Equation (5) is obviously a nonlinear SMJS. Because of the presenting of saturation for population density, it is feasible to suppose z1 (t) ∈ [−l, l], l > 0, and then the fuzzy state model can be written as follows, which is suitable for describing model system (5) as z1 (t) ∈ [−l, l]: Rule 1: If z1 (t) is M1 , then ˜ z (t) = A˜ 1 (rt ) z (t) + B˜ 1 (rt ) u˜ (t) + B˜ w,1 (rt ) w E˙ ˜ (t). ˜ z (t) = A˜ 2 (rt ) z (t) + B˜ 2 (rt ) u˜ (t) + B˜ w,2 (rt ) w˜ (t) E˙
z (t) = z1 (t)
z2 (t)
z3 (t)
T
⎡
,
⎡
ηc − αβ γ2 − ρrt + ηl ⎢ A˜ 1 (rt ) = ⎣ β ρrt αβ γ2 − γ1 − β − ⎡ αβ − γ2 − ρηcr − ηl t ⎢ A˜ 2 (rt ) = ⎣ β ρrt αβ γ2 − γ1 − β −
ηc ρrt
ηc ρrt
α −γ2 0
⎤ 0 0⎦ 0 ⎤ c − ρr + l t ⎥ 0 ⎦ −ρrt l ⎤ − ρcr − l t ⎥ 0 ⎦
1 E˜ = ⎣ 0 0
α −γ2 0
0 1 0
ρrt l
B˜ 1 (rt ) = B˜ 2 (rt ) = [ 0 0 1 ]T , B˜ w,1 (rt ) = B˜ w,2 (rt ) =
2 ˜ [ b11 b12 b13 ]T , = 1, λ˜ 1 (z1 (t)) = i=1 λi (z1 (t)) 1/2(1 − (z1 (t)/l)), λ˜ 2 (z1 (t)) = 1/2(1 + (z1 (t)/l)). By fuzzy blending, the overall fuzzy model is inferred as follows: ˜ z (t) = E˙
2
λ˜ i (z1 (t)) A˜ i (rt ) z (t) + B˜ i (rt ) u˜ (t)
i=1
k
(6)
λi (ξ (t)) (Ai (rt ) x (t) + Bi (rt ) u (t)
i=1
+ Bw,i (rt ) w (t)
wT (t) w (t) dt < d2 , d > 0.
(8)
0
First of all, some definitions will be introduced for the development of main results throughout this paper. To this end, consider the following SMJS: (9)
Definition 1 [29]: 1) The system (9) is said to be regular in the time interval [0, T], if the characteristic polynomial det (sE − A (rt )) is not identically zero for all t ∈ [0, T]. 2) The system (9) is said to be impulse free in the interval [0, T], if deg (det (sE − A (rt ))) = rankE for all t ∈ [0, T]. Definition 2 (SSFTB) [30], [31]: The system (9) which satisfies (8) is said to be SSFTB with respect to c1 , c2 , T, Rrt , d , if the system is regular and impulse free in the time interval [0, T], and satisfies E xT (0) ET Rrt Ex (0) ≤ c21 ⇒ E xT (t) ET Rrt Ex (t) ≤ c22 , ∀t ∈ [0, T] (10) where 0 < c1 < c2 and Rrt > 0. Remark 1: The regularity and nonimpulse of (E, A(rt )) ensure the existence and uniqueness of impulse free solution of system (9) in the time interval [0, T]. Lemma 1 [32]: If the following conditions hold: Mii < 0, 1 ≤ i ≤ r, 1 1 Mii + Mij + Mji < 0, 1 ≤ i = j ≤ r r−1 2 r r
In the following, the related definitions and results will be introduced for singular Markovian jump fuzzy systems (SMJFSs) of the general form firstly: E˙x (t) =
T
then the following parameterized matrix inequality holds:
+ B˜ w,i (rt ) w˜ (t) .
of If-Then rules, Mij (i ∈ T = {1, 2, . . . , k}, j = 1, 2, . . . , ¯l) are fuzzy sets, ξ1 (t), ξ2 (t), . . . , ξ¯l (t) are premise variables. E ∈ Rn×n with rankE = r < n. Ai (rt ), Bi (rt ), Bw,i (rt ) are real known matrices with appropriate dimensions. ξ(t) = ¯ [ξ1 (t) ξ2 (t) · · · ξ¯l (t)]T , βi (ξ(t)) = lj=1 Mij (ξj ) is the membership function of the system with respect to the ith plant rule. Letting λi (ξ(t)) = βi (ξ(t)) k βi (ξ (t)), it follows that i=1
k λi (ξ (t)) ≥ 0, i=1 λi (ξ (t)) = 1. The Markovian process {rt , t ≥ 0} is the same as in (3). Moreover, the disturbance w (t) ∈ Rq satisfies
E˙x (t) = A (rt ) x (t) + Bw (rt ) w (t).
Rule 2: If z1 (t) is M2 , then
where
3
(7)
where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input, w(t) ∈ Rq is the external disq turbance which belongs to L2 [0, ∞), k is the number
αi (t)αj (t)Mij < 0
i=1 j=1
where αi (t) ≥ 0 and ri=1 αi (t) = 1. Lemma 2 [33]: Given matrices E, X > 0, Y, if ET X + Y T is nonsingular, there exist matrices S > 0, L, such that ES + LT = (ET X + Y T )−1 , where X, S ∈ Rn×n , Y, L ∈ Rn×(n−r) , and , ∈ Rn×(n−r) are any matrices with full column satisfying ET = 0, E = 0. Remark 2: In [34, Lemma 3], when P is a positive definite matrix and EP + UST V T is nonsingular (these two conditions can usually be concluded from theorem’s conditions), it can be seen as a special case of Lemma 2 here by letting V S˜ = Y and UST = L.
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For notational simplicity, in the sequel, for each possible p = rt ∈ S, i ∈ T , Ai (rt ), Bw,i (rt ) are respectively, denoted by Api , Bw,pi , and so on.
where
p,ij ∗
XpT Bw,pi Xp = (ET Pp + Sp RT )T , p,ij = −Q2p
N T BT )X + X T A + B K T (ATpi + Kpj pi pi pj + p q=1 πpq E Pq E + pi p
p,ij =
T M K − τ ET P E, R ∈ Rn×(n−r) is any matrix with M1p + Kpj 2p pj p
III. F UZZY O PTIMAL G UARANTEED C OST C ONTROL In this section, the design of a mode-dependent control law for system (7) which contains the bio-economic SMJFS (6) as its special case will be focused on such that the closed loop system is SSFTB with respect to c1 , c2 , T, Rrt , d and the following cost function of the system satisfies certain requirement: ⎫ ⎧ T ⎬ ⎨ JT (rt ) = E xT (t) M1rt x (t) + uT (t) M2rt u (t) dt ⎭ ⎩
full column satisfying ET R = 0. And the upper guaranteed cost bound can been chosen as ! 2 2 τT max λmax Q1p c1 + max λmax Q2p d . ψ0 = e p∈S
p∈S
Proof: Firstly, it can be proved that the singular system (13) is regular and impulse free in the time interval [0, T]. By Lemma 1, from (14) and (15) k k
0
λi (ξ (t)) λj (ξ (t)) p,ij < 0.
(18)
i=1 j=1
(11) From (18) where M1rt and M2rt are two given symmetric positive definite matrices for each rt ∈ S. 3: For the bio-economic SMJFS (6), Remark T E{ 0 zT (t)M1rt z(t)dt} represents the average cost per unit volume to maintain the biomass in the time interval T [0, T], and E{ 0 uT (t)M2rt u(t)dt} represents the average cost per unit volume to control the biomass in the time interval [0, T]. Based on the parallel distributed compensation, the following state feedback fuzzy control law is employed: up (t) =
k
k k
λi (ξ (t)) λj (ξ (t))
i=1 j=1
+ XpT Api + Bpi Kpj + πpp − τ ET Pp E < 0.
Since rankE = r < n, there exist nonsingular matrices G and H such that 0 I GEH = r 0 0 then R can be rewritten as
λi (ξ (t))Kpi x (t)
(12)
where Kpi (i ∈ T ) are the local control gains for p ∈ S. The objective is to determine the feedback gains Kpi , p ∈ S, i ∈ T , such that the closed-loop system k k
λi (ξ (t)) λj (ξ (t)) Api + Bpi Kpj x (t)
i=1 j=1
+ Bw,pi w (t)
(13)
is SSFTB with respect to c1 , c2 , T, Rrt , d and the cost function (11) has an upper bound. system (13) is SSFTB with respect to Theorem 1: The c1 , c2 , T, Rrt , d and the cost function (11) has an upper bound in the time interval [0, T], if there exist a scalar τ ≥ 0, matrices Pp > 0, Q1p > 0, Q2p > 0, and Sp , 1 ≤ i = j ≤ k, p ∈ S, such that p,ii < 0 1 1 p,ij + p,ji < 0 p,ii + k−1 2 1
(14) (15)
1
ET Xp = ET Rp 2 Q1p Rp 2 E max λmax Q1p c21 + max λmax Q2p d2 p∈S p∈S < min λmin Q1p c22 e−τ T p∈S
(16)
(17)
R = GT
i=1
E˙x (t) =
T T Bpi Xp ATpi + Kpj
0 ¯
¯ ∈ R(n−r)×(n−r) is any nonsingular matrix. Denote where Pp1 Pp2 G−T Pp G−1 = Pp3 Pp4 S H T Sp = p1 Sp2 G
k k
λi (ξ (t)) λj (ξ (t)) Api + Bpi Kpj H
i=1 j=1
=
Ap1 (t) Ap3 (t)
Ap2 (t) Ap4 (t)
for every p ∈ S. Pre and postmultiplying (18) by H T and H, it follows that for each p ∈ S: T ¯ p2 ¯ T Ap4 (t) < 0 ATp4 (t)S + Sp2
which implies Ap4 (t) is nonsingular. Thus, for each
p ∈ S, det(sE − ki=1 λi (ξ(t))Api ) is not identity zero and
k deg(det(sE − i=1 λi (ξ(t))Api )) = rankE. According to Definition 1, the system (13) is regular and impulse free in the time interval [0, T]. Consider the Lyapunov functional candidate as V (x (t) , p) = xT (t) ET Pp Ex (t)
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for the system (13). Let L be the weak infinitesimal generator along the solution of (13), then, for each p ∈ S
LV (x (t) , p) = 2xT (t) ET Pp + Sp RT E˙x (t) ⎞ ⎛ N + xT (t) ⎝ πpq ET Pq E⎠ x (t)
Therefore, it follows that condition (17) implies: E xT (t) ET Rp Ex (t) ≤ c22 for all t ∈ [0, T]. Once again from (18) and (19), it can be easily seen LV (x (t) , p) < τ V (x (t) , p) + wT (t) Q2p w (t) − xT (t) M1p x (t) + uT (t) M2p u (t) . (23)
q=1
⎞ ⎛ k k = ηT (t) ⎝ λi (ξ (t)) λj (ξ (t))1p,ij ⎠ η (t)
Further, (23) can be represented as L e−τ t V (x (t) , p) < e−τ t wT (t) Q2p w (t) − e−τ t xT (t) M1p x (t) + uT (t) M2p u (t) .
i=1 j=1
(19)
where η(t)
=
p,ij =
[ xT (t) N
q=1
wT (t) ]T , 1p,ij
=
p,ij ∗
XpT Bw,pi −Q2p
LV (x (t) , p) < τ V (x (t) , p) + wT (t) Q2p w (t). Further, (20) can be rewritten as L e−τ t V (x (t) , p) < e−τ t wT (t) Q2p w (t).
(20)
(21)
Integrating (21) from 0 to t, it can be obtained e
E {V (x (t) , p)} < E {V (x (0) , r0 )} t + E e−τ t wT (t) Q2p w (t) dτ. (22)
T
Noting that τ ≥ 0, t ∈ [0, T] and condition (16) E xT (t) ET Pp Ex (t) = E {V (x (t) , p)} t < eτ t E {V (x (0) , r0 )} + eτ t e−τ θ wT (θ ) Q2p w (θ ) dθ 0 ! 2 2 τt ≤ e max λmax Q1p c1 + max λmax Q2p d . p∈S
p∈S
Taking into account that E xT (t) ET Pp Ex (t) & =E
1 1 xT (t) ET Rp2 Q1p Rp2 Ex (t)
0
T
